Limnol. Oceanogr., 51(5), 2006, 1956–1968
2006, by the American Society of Limnology and Oceanography, Inc.
E
Boundary layer turbulence and flow structure over a fringing coral reef
Matthew A. Reidenbach1 and Stephen G. Monismith2
Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, California 94305-4020
Jeffrey R. Koseff
Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, California 94305-4020
Gitai Yahel 3 and Amatzia Genin
The Hebrew University of Jerusalem, H. Steinitz Marine Biological Laboratory, The Interuniversity Institute for Marine
Sciences, P.O. Box 469, Eilat 88103, Israel
Abstract
Measurements of velocity and rates of turbulence were made across a fringing coral reef in the Gulf of Aqaba,
Red Sea, to determine the effect that the rough topography has on boundary layer mixing and flow dynamics.
Observations were made at two fore-reef sites and a nearby sandy slope. The friction velocity, u*, and drag
coefficient, CD, were determined directly from turbulent Reynolds stresses measured using acoustic Doppler
velocimeters. Values of CD for the coral substrates ranged from 0.009 to 0.015, three to five times greater than
over the sandy bottom site. The turbulence dissipation rate, e, was determined by fitting spectra of vertical
velocity to the theoretical ‘‘5/3’’ law expected for the inertial subrange of turbulence. There was a local balance
between production and dissipation of turbulent kinetic energy, signifying that we could estimate u* from either
the mean velocity profile, turbulence, or dissipation rate of turbulent kinetic energy. Estimates from all three
measures agreed well with mean u*/Uo ranging from 0.10 6 0.03 to 0.12 6 0.03, indicating that existing turbulent
boundary layer flow theory can be applied to flows over the rough topography of coral reefs. The bottom
topography, by enhancing both reef scale and local drag and mixing levels, allows reef biota to more effectively
exchange dissolved and particulate matter with oceanic waters.
Many coral-reef organisms have limited or no mobility
and thus depend on water motion for their basic functions,
including food delivery (Kiflawi and Genin 1997; Sebens et
al. 1997), production (Koehl and Alberte 1988; Denisson
and Barnes 1988), respiration (Goldshmid et al. 2004),
uptake of nutrients (Patterson et al. 1991; Atkinson et al.
2001), and larval dispersal (Harii and Kayanne 2003;
Sponaugle et al. 2005). The relevant hydrodynamics range
from large-scale circulation patterns that determine rates of
dispersal across the reef (Hamner and Wolanksi 1988)
through microscale turbulence, e.g., motions with length
scales near the Kolmogorov scale (Tennekes and Lumley
1972), that influence particle capture and mass flux on the
scale of an individual coral polyp (Sebens 1997; Helmuth et
1 Present address: Department of Integrative Biology, University of California, Berkeley, California 94720.
2 Corresponding author.
3 Present address: Department of Biology, University of
Victoria, P.O. Box 3020 Stn CSC, Victoria, British Columbia,
V8W 3N5 Canada.
Acknowledgments
We thank Moty Ohevia, Shiri Eckstein, Ruty Motro, Inbal
Ayalon, Eaton Dunkelberger, Roi Holtzman, and Susanne
Neilsen for their technical support to the field program and
Derek Fong for his assistance in bathymetry mapping.
This work was funded by the National Science Foundation
grant OCE 0117859, US-Israel Binational Science Foundation
grant 1997450, and the Stanford Bio-X Interdisciplinary Initiatives program. M. Reidenbach was also supported by a Stanford
Graduate Fellowship.
al. 1997). The link between these scales is the bottom
friction, i.e., shear stresses that develop because of flow
over the reef. At intermediate scales of water motion over
centimeters to meters, dispersal processes and food
acquisition depend on how near-bed flow dynamics
exchange mass between the reef and the surrounding
ocean. For example, the boundary layer within a few
meters of the reef is sometimes severely depleted of
planktonic food (Yahel et al. 1998, 2005) because of
grazing by the reef community. Replenishment of these
food sources depends upon turbulent mixing near the reef
(Genin et al. 2002).
Although local, instantaneous turbulent events are
important to benthic communities (Abelson and Denny
1997; Crimaldi et al. 2002), the temporally and spatially
integrated structure of the boundary layer often best
describes relations between flow and transport processes.
For instance, it has been shown that the uptake of nutrients
by reefs is proportional to the friction velocity, u* (Baird
and Atkinson 1997; Falter et al. 2004). In the same way, it
is known that turbulent mixing, represented by the nearbed eddy diffusivity, plays a key role in determining the
strength of concentration boundary layers that develop
because of benthic grazing (O’Riordan et al. 1993). The
average bed stress also plays a critical role in controlling
exchange between the reef and the open ocean (Hearn 1999;
Monismith et al. 2006). Therefore, an understanding of the
structure and dynamics of boundary layer flows and how it
relates to smaller- and larger-scale flow phenomena offers
valuable insight into the functioning of coral reefs.
1956
Turbulence over a coral reef
1957
Fig. 1. (a) Site map of the Gulf of Aqaba, with the location of the study area marked with
a filled square. (b) Bathymetry map of the three study sites: A, deep coral site (13 m); B, shallow
coral site (9 m); and C, sandy bottom site (11 m). Depth contours are in 4-m increments with
meter markings shown.
Turbulent boundary layers have been investigated in
a variety of coastal environments where the bottom is
nearly smooth (Gross and Nowell 1983; Sanford and Lien
1999; Stacey et al. 1999a). In these cases, flow structure
generally has adhered to the ‘‘law of the wall’’ for turbulent
boundary layers, i.e., there is a region near the boundary in
which the velocity varies logarithmically and the dissipation of turbulent kinetic energy (TKE), e, decreases like z21
away from the boundary. Under steady, uniform flow this
model can then be used to estimate u* and the coefficient of
drag, CD (Dewey and Crawford 1988; Kim et al. 2000).
From this, parameters such as turbulence length scales and
mixing rates can be calculated. These measurements,
however, are made under the assumption that there is an
equilibrium in which production of TKE, P, is in local
balance with e (Lu et al. 2000). However, direct measurements of both production and dissipation are rare, fairly
recent (Krogstad and Antonia 1999; Shaw et al. 2001), and
only observed over relatively smooth channel bottoms.
Few studies have looked at flow over extremely rough
bottom topographies such as coral reefs. In this environment, assumptions of quasisteady, uniform flow may not
hold because the roughness is often patchy, with reef area
intermixed with regions of sandy bottom. For example,
over a reef system in the Caribbean, Lugo-Fernandez et al.
(1998) inferred drag coefficients between 0.06 and 0.2 (see
also Hearn 1999). These values are large compared to the
canonical value of 0.003 measured over sand or mud
bottoms (Heathershaw and Simpson 1978). Whereas this
past work suggests that a much higher drag is associated
with coral reefs, no detailed studies have been made of how
this drag relates to mixing within the benthic boundary
layer.
The main goals of this study were therefore (1) to
describe the structure of turbulence and flow properties
over the reef; (2) to examine whether the turbulent
boundary layer obeys the wall-layer scaling, as is found
for smoother boundary layers; and (3) to determine rates
of production and dissipation of TKE and assess whether
the assumption of local equilibrium holds for the reef
environment. In pursuing these objectives, key information
required for realistic modeling of turbulent flow dynamics
in coral reefs was obtained.
Materials and methods
Study site—The experiments were conducted over the
fringing coral reefs in the Gulf of Aqaba, Red Sea, between
29 August and 14 September 1999. The study site (Fig. 1),
located offshore of the H. Steinitz Marine Biology
Laboratory, Eilat Israel, is part of a 5-km stretch of
coastline containing reef interspersed with open sand
regions devoid of corals. The reef is dominated by
hermatypic corals. Soft corals, anemones, sponges, tunicates, and polychaetes are also common (Yahel et al. 2002
and references therein). The fringing reef begins at a water
depth just below the mean low tide mark, extending to 80m depth, although some corals are found as deep as 150 m
(Fricke and Schuhmacher 1983). The average slope of the
bathymetry between 0 m and 50 m is 24u, with two mildly
1958
Reidenbach et al.
Fig. 2. (a) Coral site A with an acoustic doppler current
profiler in the foreground and the acoustic Doppler velocimeters
array in the background, (b) site A with plastic sheeting covering
the bottom topography, and (c) sandy Bottom site C.
sloped terraces at depths of 8 m and 24 m. The percent
cover of live corals during the time of this study was 14%;
rocky substratum, 29%; and sandy substrate, 57%.
Two sections of the coral reef (A and B in Fig. 1b) and
a single sandy site (C in Fig 1b) were selected for the
experiment. The two coral sites were at mean depths of
13 m and 9 m, where the bottom slopes were 27u and 18u,
respectively, whereas the sandy site was at 11-m depth with
a bottom slope of 24u. Roughness transects were conducted
over the reef sites to determine the typical topographic
roughness height of the coral community. Approximate
roughness lengths were measured by stretching a taught
line connected to two buoys across 15 m in the along-shore
direction of the test section. The line was positioned 1 m
above the mean bed height, and the roughness was
estimated by measuring the vertical distance, to the nearest
half centimeter, between the line and the highest point of
relief. These measurements were repeated at 0.5-m increments along the line. Five transects moving from onshore
to offshore were completed, which gave roughly 150
measurements at each site. For both coral sites, the mean
height of the roughness was 20 cm with a standard
deviation of 16 cm. Measurements at the sandy site were
not taken, but roughness elements (primarily gravel and
small rocks) were visually estimated to be 2–3 cm. This site
was located in the middle of an approximately 100-m long
section of a smooth, sandy, and loose-gravel bottom with
rare occurrence of corals (,1% cover). To further evaluate
the contribution of small-scale roughness to the development of flow and turbulence characteristics and to serve as
a biological control for grazing in a companion study
(Genin et al. 2002), clear plastic sheeting was suspended
just above a 20 3 10 m area of the reef at site A. This
sheeting effectively smoothed the roughness of individual
coral colonies while retaining the general shape of knolls,
clumps of corals, and the rocky substrate (Fig. 2). All sites
were ,75 m offshore, which enabled instruments to be
cabled directly to computers within the laboratory, located
25 m from the shoreline.
The 17-day-long summer experiment began just as air
and water temperatures started to decline, following a peak
in air temperatures of . 44uC. Water temperature steadily
decreased from 28uC at the start of the experiment to
25.5uC at its end, with a concurrent drop in salinity from
40.8 to 40.6 (Fig. 3). Tidal variations in surface-water level
were approximately 0.8 m. Winds during the experiment
were always to the southwest with a mean velocity
fluctuating diurnally between 4 m s21 during the night
and 7 m s21 during the day. A common feature of this reef,
as is the case in many fringing and lagoonal reefs, was
the absence of wave activity. Because of the short
fetch between the northern end of the gulf and the study
site, surface waves during the experiment were minimal,
with wave heights ,20 cm. No oscillatory motion was
measured at depth at any of the study sites. Daily
conductivity and temperature profiles (taken with an
Applied Microsystems conductivity, temperature depth
[CTD] probe) indicated that the thickness of the upper
mixed layer was between 15 m and 25 m. For a few hours
during the mid-afternoon, solar heating created a weak,
near-surface stratified layer 1–2-m thick, which was mixed
out during the cooler nighttime hours (Monismith et al.
2006). Conductivity and temperature profiles showed no
measurable stratification between 2 m below the surface
layer and the bottom, with the gradient Richardson
number well below the critical value of 0.25. Detailed
oceanographic conditions of the region are described
further in Reiss and Hottinger (1984). Measurements at
site A were made during days-of-the-year 238–244 (coinciding with spring tides), at site B during days 244–250
(neap tides), and at the sandy site C during days 250–255
(spring tides). Time is reported as local (GMT + 2 hours).
Turbulence over a coral reef
1959
Fig. 3. Time series of (a) sea surface elevation, (b) surface water temperature, (c) salinity,
(d) wind magnitude, and (e) wind direction in degrees from true north. All parameters were
measured from the laboratory pier, located 100 m to the north of the coral study sites.
Observational setup—Three acoustic Doppler velocimeters ([ADV] 10-MHz Ocean ADV, Sontek Inc.) measured velocities at a rate of 25 Hz at z 5 0.1, 0.3, and 1.0 m
above the bottom. The ADVs were located within a uniform
patch of coral along the center of the reef. Located 5 m
from the array was an acoustic Doppler current profiler
([ADCP] WH-600 ADCP, 600 KHz, RDI Inc.) programmed to profile the water column in 0.5-m bins
between 1.5 m above bottom and 1 m below the water
surface. The ADCP recorded 100-ping velocity ensembles
every 40 s, giving a per sample standard error of 1 cm s21.
All instruments were placed as close as possible to one
another to ensure they measured the same flow conditions,
but were arranged perpendicular to the main axis of flow to
minimize flow disturbances by neighboring instruments.
Measurements were made relative to datum equal to the
local height of the sand bed at the base of the corals. This
allowed for consistent measurements between both the
coral sites and the sand bottom site and enabled us to
determine the relative effect of the coral roughness on the
structure of the boundary layer.
Data analysis—Instantaneous velocity measurements
were rotated into local bathymetry coordinates of alongshore (U), across-shore (V), and vertical (W) directions.
Since most of the water motion was driven by tidal action
in the along-shore, this direction (45u from true N during
positive flow, 225u during negative flow) was used as the
main axis of flow. Mean velocity and turbulence statistics
were computed by averaging over 10-min intervals. This
time period was chosen because in statistical tests,
10 minutes often emerges as the best balance between
obtaining convergence of the mean statistics while minimizing velocity drift caused by changes in tidal flow
conditions (i.e., Gross and Nowell 1983). The ADV records
were combined with ADCP records to make complete
velocity profiles throughout the water column. For the
turbulence measurements, which were all measured within
1 m from the bottom, a coordinate system was used in
which z is perpendicular to the bottom. Above 1 m,
velocity statistics were calculated using a reference frame
where z is vertical since it was assumed that flow properties
were less influenced by the bottom slope higher within the
water column. This change in coordinate system had no
effect on the velocity component in the along-shore flow
direction and, on average, there was ,10% variation in the
turbulence measurements (u0 w0 ) calculated using the two
reference frames. The following conventions were used for
flow direction: positive along-shore flows correspond to
a Northeast-ward (NE) current, whereas negative values
correspond to a Southwest-ward (SW) current. Positive
across-shore flow is offshore (to the Southeast at 135u), and
positive vertical velocities represent upwelling.
Bottom shear stress, to ~ru2 (where r is the density of
the fluid) is arguably the single-most important boundary
layer parameter because it is a measure of the frictional
interaction between the flow and the bottom. However,
direct measurements are difficult in the field because of
complex bed geometries. Instead, mean flow and turbu-
1960
Reidenbach et al.
stress region, Reynolds stresses can be used to estimate the
friction velocity as
1
ð2Þ
u ~u0 w0 2
Velocity fluctuations were calculated by subtracting mean
velocity values over 10-min intervals from instantaneous
measurements u9 5 u 2 U, v9 5 v 2 V, w9 5 w 2 W. Using
the three ADVs sampling at 25 Hz, direct calculation of
Reynolds stresses were made, and their average stress was
used in Eq. 2.
Dissipation technique: The rate P at which energy is
produced from mean flow into TKE is estimated as the
product of the Reynolds stress times the mean shear:
P~{u0 w0
Fig. 4. Typical power spectrum using the vertical velocity
component at z 5 1 m (site B). The ADV noise floor is
0.005 cm2 s21. Note no existence of measurable wave energy
within the spectrum.
lence measurements were obtained higher in the water
column, and these values were used to infer stresses
imposed at the bed. Three common techniques were used
to estimate u*: (1) the logarithmic law, (2) the covariance,
and (3) the dissipation technique.
Logarithmic law technique: For a well developed
turbulent boundary layer, an inertial sublayer region exists
where mean velocities exhibit a logarithmic profile. Within
this region, the mean velocity profile is related to the
generation of turbulence by shear at the bed, and the law
of the wall takes the form (Kundu 1990):
u ðtÞ
z{d
ln
ð1Þ
Uðz,tÞ~
k
zo
where k 5 0.41 is von Karman’s constant, d is the
displacement height, and zo is the bottom roughness length
scale. Values of u*, d, and zo were adjusted to obtain a best
fit, in the least squares sense, through the logarithm of
mean velocities. Eight measurement points were used in the
fit, three from ADV measurements between 0.1 m and 1 m
above bottom and five from ADCP measurements between
1.5 m and 3.5 m above bottom. Above this, a deviation
from a logarithmic profile occurred because of wind stress
at the surface. At each site, it was assumed that both d and
zo are properties of the bed roughness and independent of
time, which is valid for the timescales of the experiment
(Cheng et al. 1999; Stacey et al. 1999b). To ensure a well
defined log region over the measurement points used in the
fit, only profiles having an r2 . 0.8 were used in the
analysis. This gives a 95% confidence interval for u* of
6 28% (Drake et al. 1992).
Covariance technique: The estimation of u* through the
covariance technique assumes that Reynolds stresses are
constant within the inertial sublayer (Tennekes and Lumley
1972) and outside of this region linearly decrease to zero at
the outer edge of the boundary layer. Within the constant
LU
LV
{v0 w0
Lz
Lz
ð3Þ
If a balance between P and the rate of TKE dissipation (e)
occurs where P 5 e, the law of the wall in the constant
stress region can be used to relate the friction velocity to the
u3
dissipation rate as ~e (Dewey and Crawford 1988).
kz
Dissipation itself can be calculated independently through
spectral measurements of velocity fluctuations and an
equation for the friction velocity can be written as
u ~ðekzÞ
1
3
ð4Þ
Dissipation was estimated from a one-dimensional spectral
equation of vertical velocity fluctuations (Shaw et al. 2001):
9 4{ cos h
2
5
ð5Þ
Sww ðkÞ~
ae3 k{3
55
3
where Sww(k1) is the power spectral density as a function of
the wave number k; h 5 90u is the angle from the direction
of mean flow; and a 5 1.56 is the empirical Kolmogorov
constant for velocity. Using Taylor’s frozen turbulence
hypothesis, wave number space was converted to frequency
Uk1
. Spectra were formed by averaging 7
space as f~
2p
independent spectra of 2,048 sample records spanning a 10min time period (Fig. 4). To determine the location of an
inertial subrange, i.e., where spectra show a 25/3 slope, the
noise floor of the spectrum was first removed, and a linear
regression of log(Sww) versus log(f ) was carried out over n
5 148 points (spanning 0.8 decades). e was then estimated
through Eq. 5 by determining the spectral intensity
[f1, Sww(f1)] within the inertial subrange, giving 95%
confidence of 6 0.13e. Most of the uncertainty in e is in
estimating the magnitude of the power spectrum at a given
frequency (Gross and Nowell 1985). To ensure a full
separation of scales, a minimum shear Reynolds number,
u kz
criterion was applied to the data (Gross and
Re ~
n
Nowell 1985). Although there is no universal value of Re*
above which a well defined inertial subrange exists, our
Turbulence over a coral reef
1961
Fig. 5. Combined ADCP and ADV mean velocities at site B in the along-shore, acrossshore, and vertical directions. The ADVs were turned off between days 248.3 to 248.5 to backup
the data, therefore there is no flow information for this time period within 1 m from the bottom.
data showed that a definitive 25/3 slope region existed in
spectra when u w0:4 cm s{1 . This criteria gives a minimum
Re* 5 2,000 at z 5 1 m, similar to the criteria of Re* 5
2,500 found by Huntley (1988).
Results
Mean currents and Reynolds stresses—Flows along the
reef were primarily driven by semidiurnal internal tides
(Monismith and Genin 2004). Mean currents at the coral
sites were 5 cm s21 with peak flows on the order of
20 cm s21 (Fig. 5). Velocity profiles as well as daily CTD
profiles indicated that the bottom boundary layer had
a thickness of 5 m to 7 m and was not influenced by
stratification at the study sites. Because of a wind stress
imposed at the surface, there was often a surface boundary
layer that blended in with the bottom boundary layer at
mid-water column. This caused the bottom boundary
layer to deviate from a logarithmic profile starting at
a height of 3.5 m to 5 m above the bed. Overall, the winddriven flow led to stronger currents to the SW than to
the NE. As described in Monismith et al. (2006), at all
sites there were across-shore flows associated with
diurnal cooling and heating of waters over the reef
(Fig. 5b).
The alongshore Reynolds stress tensor, i.e., u0 w0 , and the
1
TKE, q2 ~ u0 u0 zv0 v0 zw0 w0 , for all sites are shown in
2
Fig. 6. The Reynolds stress (u0 w0 ) exhibited expected
boundary layer characteristics, with values at the three
ADV measurement heights being roughly equal. This
agreement suggests the existence of a constant stress region
within the near-bed turbulent boundary layer. TKE at all
the sites decreased with increasing height from the bed.
Mean estimates of {u0 w0 =q2 , which is the amount of
frictional energy imparted at the boundary due to TKE,
were 0.12, 0.11, and 0.11 (60.02) at coral sites A and B and
sand site C, respectively. These values are consistent with
estimates found in other tidally dominated flows of 0.08
(Lu et al. 2000) and 0.14 (Gross and Nowell 1983).
Between days 242.5 and 244.5, the plastic sheeting placed
over a 20 m 3 10 m section of reef at site A removed the
fine-scale relief and small-scale roughness of the coral
benthic community. Nonetheless, there was no statistical
difference in measured turbulence levels with
and
without
the sheeting in place (site averaged u0 w0 ~0.59 6
0.14 cm2 s22 without cover and 0.52 6 0.12 cm2 s22 with
cover). Even 10 cm above the bed, the integrated effect of
the underlying reef topography appears to determine
turbulence statistics, not the drag of individual coral heads
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Reidenbach et al.
Fig. 6. (a) Mean velocities, (b) Reynolds stresses, and (c) TKE for z 5 1 m, 0.3 m, and 0.1 m
for sites A, B, and C. The time period during which site A was covered with plastic sheeting is
marked in (a).
or the localized friction around the coral fine-scale
branching structure.
Production and dissipation of turbulent kinetic energy—
TKE production estimates were obtained as the product of
the ADV-derived Reynolds stresses and the local velocity
gradients at the ADV heights, obtained from logarithmic
fits to the mean velocity profile. Figure 7 shows the
comparison between production and dissipation estimates
at z 5 0.1 m, 0.3 m, and 1.0 m for coral sites A and B and
the sand site C. The plots indicate variability during weak
flow conditions, but values of production and dissipation
converge at higher intensities (r2 for the P 5 e fit at z 5
1.0 m, 0.3 m, and 0.1 m are site A, 0.54, 0.55, 0.60; site B,
0.69, 0.84, 0.71; and site C, 0.54, 0.73, 0.82; respectively).
Most of the observed e values lie between 1023 and
1 cm2 s23 (equivalent to 1027 and 1024 W kg21). Systematic differences in production and dissipation levels
occurred at values below 1022 cm2 s23, suggesting a noise
floor for P and e estimates.
As the strength of the mean flow increases, production
and dissipation levels also increase and a local production–
dissipation balance is evident. This is consistent with
expectations for unstratified near-wall regions in a turbulent
boundary layer. During periods of high flow, near-bed
maximum P and e rates were of similar magnitude at both
the coral and sand sites, i.e., on the order of 1 cm2 s23.
Given that peak velocities at the coral sites were half those
at the sand site, it is clear that the flow over the reef is
substantially more turbulent at a given velocity than flow
over the sandy bed, i.e., it has relatively higher dissipation
rates and turbulence intensities than does the sandy site.
TKE production decayed away from the boundary, and
to a first approximation, follows the relation P , 1/z that is
expected from the law of the wall. Dissipation profiles also
exhibited this trend, especially during high flow periods
when a well defined inertial range existed. This balance also
signifies that non-local vertical transport of TKE, which
often occurs within the wakes of blunt bodies and causes an
imbalance between P and e (Finnigan 2000), was negligible.
Measurements of turbulent transport (calculated as
Lq0 2 w0
) within 1 m of the reef was generally on the order
Lz
of 1024 cm2 s23, which was 102–104 times lower than local
production rates.
Roughness length scales and friction velocities—Using
a least-squares regression, values of the displacement
heights converged to d 5 0.06 6 0.02 m for each of the
coral sites and d 5 0.02 6 0.005 m for the sand site. Fixed
values of zo, termed zo-global, were calculated as the logaverage of the individual 10-min estimates of zo (i.e.,
zo{global ~expvln(z0 )w). Variations in zo-global estimates
were found depending on the site and direction of flow,
presumably because of differences in upstream topography.
Therefore, separate estimates were formed for both NE and
SW flow conditions (Table 1). This within-site variability
was not unexpected because of the many length scales that
Turbulence over a coral reef
1963
Fig. 7. Production versus dissipation at the three ADV locations z 5 1.0 m, 0.3 m, and
0.1 m, respectively.
Table 1. Global roughness lengths for flows to the NE and
SW. Uncertainties are reported as 95% confidence on the mean.
Site
NE zo-global (cm)
SW zo-global (cm)
A
B
C
1.360.2
1.960.2
0.460.1
1.060.2
3.960.3
0.160.1
exist within the reef, including the size of the coral
roughness elements, their spacing, and the relative contribution their morphology plays in generating drag. Values
for the coral sites ranged from zo-global5 1.0–3.9 cm,
compared to 0.1–0.4 cm for the sand site. This 10-fold
increase in roughness length is because of the large effect
that the coral has on the flow.
Using measured values of d and zo-global, friction
velocities were estimated using the log-law technique and
compared to estimates obtained from the covariance and
dissipation techniques (Fig. 8). In general, there is very
good agreement between the techniques, and a linear
relation was found between u* and Uo for all the estimates.
This is indicative of the interdependence of turbulence,
velocity shear, and energy levels on mean flow conditions.
The log-law technique tends to slightly overpredict u*
compared to the other two techniques (Table 2). The likely
cause for this discrepancy is the existence of a pressure
gradient that acts to reduce the magnitude of the Reynolds
stress away from the bed (Gross and Nowell 1985; Rippeth
et al. 2002). Nonetheless, the estimates of u* from the three
techniques agree within 95% confidence intervals. For
a given Uo, u* across the reef sites was consistently double
that measured over the sandy bed, with mean u*/ Uo of 0.12
6 0.02 and 0.10 60 .02 at coral sites A and B, respectively,
compared to 0.06 6 0.01 for the sand site C.
Mixing coefficients and turbulence length scales—Measurements of mean velocity and turbulence parameters
enable the calculation of turbulence characteristics that are
of particular interest in transport studies: namely, the eddy
1964
Reidenbach et al.
Fig. 8. Comparison of u* estimates from log-law, covariance, and dissipation techniques for
coral site A, coral site B, and sand site C. Each estimate is the binned average over a 1 cm s21
increment of Uo. Note the doubling of the mean velocity scale of plot C compared to A and B,
indicating that the relative friction velocity to mean velocity ratio of the coral sites is twice that of
the sandy bottom site.
Table 2.
u
Estimates of U by the log-fit, covariance, and dissipation techniques during SW and NE flow conditions.
o
Site A
Site B
Site C
Technique
SW
NE
SW
NE
SW
NE
Log-fit
Covariance
Dissipation
0.1160.02
0.1060.02
0.1260.02
0.1260.03
0.1260.03
0.1160.02
0.1160.02
0.1060.02
0.1060.02
0.1060.02
0.1060.02
0.1060.02
0.0660.01
0.0560.01
0.0560.01
0.0860.01
0.0760.01
0.0660.01
Table 3.
Mean eddy viscosity (ne) and mixing length (lm) estimates at the three ADV locations.
Mixing parameter
z (cm)
Site A
Site B
Site C
ne (cm2 s21)
100
30
10
100
30
10
3567
1162
2.360.5
3863
12.760.8
3.360.3
2565
6.561.4
2.760.6
3962
10.660.5
5.260.2
3367
1162
4.160.9
4462
14.660.6
5.760.2
lm (cm)
viscosity, ne, and the mixing length, lm. The eddy viscosity is
LU
(e.g., Tennekes
defined in terms of the mean shear, S:
Lz
and Lumley 1972)
ne ~
{u0 w0
LU
Lz
ð6Þ
whereas the turbulent mixing length, which is the typical
size of a turbulent overturn, is defined as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
{u0 w0
lm ~ LU
ð7Þ
Lz
Values for these two parameters at all three sites are
summarized in Table 3. Whereas estimates of ne are
dependent on the magnitude of the flow, the mixing length
scale should be independent of flow rate. Under the
formulation of lm, if the mean shear fits the law of the
wall theory, then in unstratified, wall-bounded shear flows
the mixing length is proportional to distance from the wall,
i.e., lm ~kz. At the ADV sampling heights, 100 cm, 30 cm,
and 10 cm, this gives a prediction of lm 5 40 cm, 12 cm,
and 4 cm, respectively. Our measurements agree with these
predicted values and have similar magnitudes both at the
coral sites and the sand site. Likewise, using wall scaling in
the bottom boundary layer, ne ~u kz. Values of ne
calculated from Eq. 6 and normalized by u*kz agree; we
find ne =ðu kzÞ of 1.01 6 0.10, 0.93 6 0.08, and 1.06 6 0.08
for sites A, B, and C, respectively. This agreement is
important because it allows for the estimation of the eddy
diffusivity, ke, from the law of the wall via the Reynolds
analogy (ne %ke ), and, hence, the computation of the flux
of scalars (i.e., plankton, larvae, etc.) from measures of
concentration gradients.
Drag coefficient—The coefficient of drag, CD, is defined
as the square of the friction velocity divided by the square
of the reference velocity, Uo.
CD :
u2
U2o
ð8Þ
Turbulence over a coral reef
1965
Fig. 9. Along-shore velocity Uo jUo j versus u0 w0 for sites A (with and without covering), B,
and C. The slope of the best fit line, as reported on the graph, corresponds to the coefficient of
drag, CD.
Uo can be defined in a number of ways, but here the
reference used is the mean velocity at 1.0 m above the
bottom. Using the covariance technique to estimate u*2, CD
was calculated as the slope of the best fit line between the
square of Uo versus u0 w0 (Fig. 9). Drag values over the
coral sites were 3–5 times larger than those estimated at the
sand site, characterized by 0.002 6 0.001 , CD , 0.004 6
0.001, depending on flow direction. These measurements
are similar to the canonical value of CD 5 0.003 used for
flows over relatively smooth sand or mud beds. The
average drag coefficient across the reef was CD 5 0.011, but
depending on flow direction, values varied from 0.009 to
0.015. These variations in drag between site and direction
of flow are likely caused by the combined effect of
differences in upstream topography and changes in water
depth because of location and tidal variations.
Drag calculations using u* estimates from the logarithmic technique give similar, but slightly higher values than
those calculated through Reynolds stress estimates (CD
from log-fits of the mean velocity profile ranged from 0.010
to 0.016 over the reef). Both techniques for estimating drag
show consistent trends of reef sites having 3–5 times larger
drag than sites characterized by sand bed roughness. At site
A, both the logarithmic and covariance techniques showed
no statistical difference in drag with or without the plastic
sheeting in place (with the sheeting, values of CD 5 0.010 6
0.004 and 0.014 6 0.002 were measured for SW and NE
flows, respectively). This is likely due to the greater relative
influence of the integrated upstream roughness rather than
the effects of local topography adjacent to the sampling
location.
Discussion
Flow characteristics of the reef boundary layer indicate
that a local equilibrium exists, where the amount of
turbulent energy produced is equal to the amount
dissipated. This equilibrium is the foundation upon which
the classical law of the wall that describes canonical
turbulent boundary layers rests. Were the law of the wall
not to hold for flows over the Eilat reef, the three
independent techniques used to compute u* would not
agree. The fact that the different approaches give consistent
answers allows us to conclude that the reef boundary layer
is such a canonical turbulent boundary layer. Thus, even
though the topography of the reef is highly variable, even
a few tens of centimeters above the reef, the profile of the
boundary layer looks similar to that of smoother boundary
layers, albeit with much higher levels of shear, turbulence,
and mixing. The utility of these measurements is that by
from these spatially integrated values of u* it is possible to
estimate other known properties of these boundary layers,
e.g., eddy diffusivities that vary linearly with distance from
the wall and are proportional to u*. Practically, it also
simplifies matters that log-fits to observed mean velocity
profiles, as can be obtained with ADCPs (something that is
much easier to do than directly measuring the Reynolds
stresses), can be used to compute u*.
1966
Reidenbach et al.
Our measurements indicate that values of zo are only
a few centimeters (between 1.0 cm and 3.9 cm) whereas the
height of the coral is much larger. However, there is
nothing inconsistent with this finding since zo is a scaling
parameter that represents not just the height of the coral,
but the integrated effects of the bed geometry. For example,
for sand grains, zo is 1/30 the sand grain diameter. In our
case, this would translate to sand grains that are 0.3–1.2 m,
at least comparable to the larger features on the Eilat reef.
Additionally, the displacement height of the reef was 6 6
2 cm in height, much less than the average height of the
coral. This indicates there is still substantial flow passing
within the reef, which is undoubtedly important for
maintaining high rates of exchange of nutrients, larvae,
dissolved gases, and other constituents with the overlying
water column.
No statistically significant changes in turbulence characteristics were observed following the covering of the reef
with the plastic sheet. Not only was the drag comparable,
but with the plastic sheet, zo–global 5 1.3 6 0.2 cm and 0.9
6 0.1 cm for NE and SW flow directions, respectively.
These are statistically similar roughness estimates compared to values without the sheet (as reported in Table 1),
suggesting that measures of shear and mixing are less
influenced by the local roughness than the integrated effects
of topography over a large area. Local reef variability or
patchiness on the order of centimeters to meters may
therefore have little effect on integrated turbulence and
mixing properties. This has important implications to reefscale modeling efforts; it may be that the larger-scale
roughness characteristics set the drag coefficient and
mixing parameters, not local roughness elements. However,
within close proximity to the bed, the detailed shape of the
benthic organisms is likely to play a more important role in
small-scale turbulent and diffusive transfer processes,
which occur immediately adjacent to the coral surfaces
(Reidenbach et al. 2006). Over complex roughness like that
of a coral reef, one might expect that shear-layer vortices
might predominate, such as have been found for seagrass
beds (Cornelisen and Thomas 2004) or even much larger
roughness elements, such as buildings (Louka et al. 2000).
This suggests that our bulk measures of drag may do well in
predicting water column mixing but do not accurately
predict small-scale flow patterns along and within the
structure of individual corals.
Interestingly, elevated values of CD measured over the
fringing reef are still a factor of 10–100 less than what has
been deduced from measurements or calibration of
circulation models for flow over coral reefs like those in
Kaneohe Bay, Hawaii (Hearn 1999) or St. Croix, U.S.
Virgin Islands (Lugo-Fernandez et al. 1998). It seems
unlikely that differences in details of the bottom topography between these various reef sites can explain this large
difference in inferred values of the drag coefficient,
especially given that zo measured for the Kaneohe reef is
smaller than what we find for the Eilat reef (Lowe et al.
2005a). The most likely cause for this variation in drag is
that the field cases reported above involved flows with
energetic surface waves, whereas in the present study waves
were insignificant. The presence of waves increases the
mean stress (Falter et al. 2005), although how exactly waves
affect drag remains uncertain and is worthy of further
study, given the importance of waves to many reefs (Lowe
et al. 2005b). However, our measured values of zo are quite
close to what McDonald et al. (2006) found for a continuous bed of 15-cm high Porites compressa heads. In
contrast, extrapolating the work of Lowe et al. (2005b)
with cylinder arrays to the case of arrays of large coral
heads, ‘‘bommies,’’ that are barely submerged (e.g., as
found in the reef lagoons on the north shore of Moorea,
French Polynesia), much larger effective drag coefficients
than what we find in Eilat might be anticipated for some
reefs.
Nonetheless, the relatively high values of measured CD
undoubtedly imply enhanced turbulent mixing. This in turn
should increase the ability of the benthic community to
graze on phytoplankton (Yahel et al. 1998; Genin et al.
2002) and zooplankton (Yahel et al. 2005; Holzman et al.
2005). The factor of four increase in CD implies a two-fold
increase in rates of mixing, thus enhancing replenishment
of planktonic food to the benthic community and possibly
increasing food uptake and coral growth rates. Likewise, if
the reef’s bottom roughness and coral growth are
correlated, a positive feedback is expected to occur during
the succession of a reef. Initially, when corals are sparse
and small, CD is low and growth is expected to be low
(assuming similar levels of competition). At more mature
stages of succession, after corals have become more
abundant and larger, turbulence-enhanced growth should
further augment the bottom roughness. However, factors
that preferentially cause mortality of large corals, e.g., coral
breakage, may offset this heightened growth, thus maintaining the reef roughness at a quasisteady state.
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Received: 30 August 2005
Accepted: 29 March 2006
Amended: 20 April 2006